The Kinematics of Conical Involute Gear Hobbing

Carlo Innocenti

(This article fi rst appeared in the proceedings of IMECE2007, The technical literature contains plenty of information the 2007 ASME International Mechanical Engineering regarding the tooth fl ank geometry (Refs. 8–9) and the setting Congress and Exposition, November 11–15, 2007, Seattle, of a hobbing machine in order to generate a beveloid gear Washington, USA. It is reprinted here with permission.) (Refs. 10–15). Unfortunately, the formalism usually adopted makes determination of the hobbing parameters a rather involved process, mainly because the geometry of a beveloid Management Summary gear is customarily—though ineffi ciently—specifi ed by It is the intent of this presentation to determine all rigid-body positions of two conical involutes that resorting to the relative placement of the gear with respect mesh together, with no backlash. That information to the standard rack cutter that would generate the gear, even then serves to provide a simple, general approach in if the gear is to be generated by a different cutting tool. To arriving at two key setting parameters for a hobbing make things worse, some of the cited papers on beveloid gear machine when cutting a conical (beveloid) gear. hobbing are hard reading due to printing errors in formulae A numerical example will show application of the and fi gures. presented results in a case study scenario. Conical This paper presents an original method to compute the involute gears are commonly seen in gearboxes parameters that defi ne the relative movement of a hob with for medium-size marine applications—onboard respect to the beveloid gear being generated. Pivotal to the engines with horizontal crankshafts and slightly proposed method, together with a straightforward description sloped propeller axes—and in automatic packaging of a beveloid gear in terms of its basic geometric features, is applications to connect shafts with concurrent axes the determination of the set or relative rigid-body positions whenever the angle between these axes is very small of two tightly meshing beveloid gears. Based on this set of (a few degrees). relative positions, the paper shows how to assess the rate of change of the hob-work shaft axis distance as the hob is fed Introduction across the work, as well as the rate of the additional rotation Conical involute gears, also known as beveloid gears, of the hob relative to the gear. These parameters have to be are generalized involute gears that have the two fl anks of the entered into the controller of a CNC hobbing machine in order same tooth characterized by different base cylinder radii and to generate a given beveloid gear. different base helix angles (Refs. 1– 4). Beveloid gears can be The proposed method also can be applied to the grinding of mounted on parallel-, intersecting- or skew-axis shafts. They beveloid gears by the continuous generating grinding process. can be cheaply manufactured by resorting to the same cutting Furthermore, it can be extended to encompass the cases of machines and tools employed to generate conventional, involute swivel angle modifi cation and hob shifting during hobbing. helical gears. The only critical aspect of a beveloid gear pair, Lines of Contact i.e., the theoretical punctiform (single-point) contact between The hob that generates a beveloid gear in a hobbing the fl anks of meshing gears, can be offset by a careful choice machining process can be considered an involute gear. Most of the geometric parameters of a gear pair (Refs. 5–7). On the commonly, such a gear is of the cylindrical type, although other hand, the localized contact between beveloid gear teeth adoption of a conical involute hob would be possible too, in comes in handy should the shaft axes be subject to a moderate, principle. For this reason, the kinematics of hobbing will be relative position change in assembly or operation. presented in this paper by referring to a beveloid hob, and

0056 GEARTECHNOLOGY July 2008 www.geartechnology.com subsequently specialized to the case of a cylindrical hob. This section introduces the nomenclature adopted in the (4) paper and summarizes known results pertaining to the loci of contact of a conical involute gear set. The reader is referred Owing to Equation 4, the following equations and ensuing to References 16 and 17 for a detailed explanation of the text and ﬁ gures will refer to the normal base pitch of right-hand

reported concepts and formulas. and left-hand ﬂ anks of both gears as p1 and p–1 respectively. The tooth ﬂ anks of a conical involute gear are portions The common perpendicular to the axes of a pair of of involute helicoids. As opposed to classical helical involute meshing beveloid gears intersects the axes themselves at

gears, the two helicoids of the same tooth of a conical involute points A1 and A2 (Fig. 2). The relative position of these axes gear do not generally stem from the same base cylinder, nor continued have the same lead.

In a beveloid gear set composed of gears G1 and G2, the

axis of gear G1 is here directed in either way by unit vector

n1. The orientation of the axis of gear G2 by unit vector n2 is then so chosen as to make the right-hand fl anks of gear G come into contact with the right-hand fl anks of gear G . 1 2 �i With reference to Figure 1, the distinction between right-hand Fi and left-hand tooth fl anks is possible based on the sign of the �i,1

ensuing quantity: �i �i,-1 �i (1) �i,1

where Fi is a point on a tooth ﬂ ank of gear Gi (i=1, 2), Oi is a

point on the axis of gear Gi, and qi is the outward-pointing unit �i vector perpendicular to the tooth fl ank at Fi. A tooth fl ank is a right-hand or left-hand fl ank, depending on whether Equation 1 results in a positive or, respectively, negative quantity. (In

Figure 1, point Fi lies on a right-hand ﬂ ank of gear Gi). In the sequel, index j will be systematically used to refer to right- Figure 1—Basic dimensions of beveloid gear Gi. hand (j = + 1) or left-hand (j = −1) tooth ﬂ anks. Figure 1. Basic dimensions of beveloid gear G i.

The basic geometry of gear Gi (i=1, 2) is deﬁ ned by its

number of teeth Ni, the radii ρi,j (j = ± 1) of its base cylinders, the base helix angles β of its involute helicoids (β < i,j i,j �� π/2; βi,j > 0 for right-handed helicoids), and the base angular y� �� thickness ϕ′i of its teeth at a speciﬁ ed cross section. All these n parameters—with the exception of N —are reported in Figure � � i �� � �� 1, which also shows the involute helicoids as stretching x� �� inwards up to their base cylinders, irrespective of their actual radial extent. �� The normal base pitch is the distance between homologous involute helicoids of adjacent teeth of the same gear. A �

beveloid gear has two normal base pitches—pi,1 and pi,–1— one for right-hand ﬂ anks and one for left-hand ﬂ anks. Their expression is: ��

x� �� (2) �� where y� z� �� �� (3)

�� n� Two beveloid gears can mesh only if they have the same normal base pitches, i.e., only if the ensuing conditions are satisﬁ ed: Figure 2—The lines of contact. �� www.geartechnology.com July 2008 GEARTECHNOLOGY 0057 (8) P3–i, –1 P 3–i,1 (9)

(10) xi (11)

Pi, –1 Pi,1 The function sign(.) in Equation 10 returns the value + 1, �i,1 �i, –1 0, or –1 depending on whether its argument is positive, zero

or negative. In Figure 3, angle θi,1 is positive, whereas angle

θi,–1 is negative. Ai Due to the square root in Equation 6, two beveloid gears yi can properly mesh only if condition

�i,1 �i, –1 (12)

is satisfi ed for both values of j. In the following equations, text and fi gures Equation 12 will be supposed as holding. Figure 3—Coordinates of the extremities of the lines of contact. Thanks to Equations 5 and 6, angle θ can be expressed Figure 3 . Coordinates of the extremities of the lines of contact. i,j as: is defi ned by their mutual distance a0, together with their relative inclination α . Specifi cally, angle α is the amplitude 0 0 (13) of the virtual rotation—positive if counterclockwise—about vector (A2–A1) that would make unit vector n1 align with unit vector n . Two fi xed Cartesian reference frames W (i =1, 2) 2 i (i=1,2; h= 3 – i; j= ± 1). are then introduced with origins at points A , x axis pointing i i The z-coordinate b of point P in reference frame W towards A , and z axis directed as unit vector n . i,j i,j i 3– i i i (i=1,2; j= ± 1) is given by: For a conventional beveloid gear set composed of two b = meshing conical involute gears that revolve about their fi xed 1,j and non-parallel axes, the locus of possible points of contact (14) between the involute helicoids of right-hand (left-hand) fl anks is a line segment that has a defi nite position with respect to either of reference frames W (i=1, 2). More specifi cally, the i Based on the relative positions of reference frames W and line of contact is tangent at its ending points P and P 1 1,1 2,1 W , together with the cylindrical coordinates ρ , θ , and b of (P and P ) to the base cylinders of the right-hand (left- 2 i,j i,j i,j 1,–1 2,–1 points P with respect to W (i=1,2; j= ± 1), the length σ of the hand) fl anks of the two gears. This is true even if the actual i,j i j line of contact P P can be determined by (Refs. 16–17): tooth fl anks—being limited portions of the above-mentioned 1,j 2,j involute helicoids—touch each other along line segments that (15) are shorter than the above-mentioned lines of contact and superimposed on them. The results summarized so far are directly applicable to The cosine and sine of the angle θ that the projection of i,j a conventional beveloid gear set, i.e., to a pair of meshing vector (P –A ) on the xy-plane of reference frame W forms i,j i i beveloid gears that revolve about their rigidly connected axes. with the x-axis of W (see Fig. 3) are indirectly provided by: i They also represent a convenient starting point for determining all possible relative positions of a beveloid hob relative to the (5) beveloid gear being machined. A Backlash-Free Beveloid Gear Set Due to the single-point contact between tooth fl anks of a (6) beveloid gear set, the assortment of rigid-body positions of a beveloid hob relative to the beveloid gear being machined where cannot be confi ned to the simple infi nity of relative positions of two gears in a conventional beveloid gear set. Otherwise, (7) a gear machined by a beveloid hob would not have involute helicoidal fl anks; rather, only one curve on these fl anks would

5008 GEARTECHNOLOGY July 2008 www.geartechnology.com belong to the desired involute helicoids. �0i Therefore the double infi nity of points on a tooth fl ank of a beveloid gear machined by a beveloid hob has to be obtained �i at least as a two-parameter envelope of the positions of the Hi hob tooth fl anks. Each of these positions must correspond to a meshing confi guration—with no backlash—of the beveloid �0i hob with the fi nished beveloid gear. There is a quadruple infi nity of confi gurations of a beveloid –1 –1 gear tightly meshing with a beveloid hob (the contacts between tan ki,–1 tan ki,1 right-hand fl anks, as well as the contacts between left-hand fl anks, each diminish by one the original six degrees of freedom that possess, in principle, a freely-movable hob that does zi not touch the gear). The double infi nity of rigid-body positions Figure 4—The developed unitary-radius cylinder of gear Gi. of the hob relative to the generated gear is only a subset of Figure 4 . The developed unitary-radius cylinder of gear Gi. the quadruple-infi nity possible meshing confi gurations. This latter set of confi gurations can be found by fi rst determining Σi,1 intersecting the minimum distance segment A1A2 (Fig. 2) the simple infi nity of all relative rigid-body positions of two at a point of the base cylinder of gear Gi (i=1,2). (Equivalently, beveloid gears tightly meshing in a conventional gear set, at the reference angular position of gear Gi the projection of which is just the scope of the present section. the base helix of Σi,1 on the unitary-radius cylinder of the gear With regards to the same set of beveloid gears considered goes through point Hi.) A generic angular position of gear Gi in the previous section, let ϕ be the common-normal angular 0i is then identifi ed by the angle γ0i of the rotation that carries base thickness of a tooth of gear Gi (i=1, 2), i.e., the angular the gear from its reference angular position to the considered base thickness at the cross-section of gear G through point A i i position. Angle γ0i—positive for a counterclockwise rotation (Figs. 1 and 2). On a generic cross-section identifi ed by the with respect to unit vector ni (Fig. 2), can also be highlighted axial coordinate zi, the tooth angular base thickness ϕi′ of a on the developed unitary-radius cylinder (Fig. 4). tooth of gear G is given by i The angular positions γ01 and γ02 of two beveloid gears that mesh together with zero backlash are clearly interrelated. An (16) obvious mutual constraint is the differential condition that stems from expressing the gear ratio in terms of the number

In this equation, quantity ki,j is deﬁ ned as of teeth Ni (i=1,2) of the two gears

(17) (18)

Incidentally, ki,j can be given a geometric meaning: If the In order to fi nd a fi nite relation between γ01 and γ02, two base helix of an involute helicoid j of gear Gi is projected maneuvers are envisaged. The fi rst maneuver starts with on the unitary-radius cylinder coaxial with the gear, then the the fi rst gear at position γ01= 0 and—by exploiting the tooth resulting projection is a helix whose inclination angle with contact between right-hand fl anks only—carries the second –1 respect to the gear axis is tan ki,j. gear at position γ02= 0. The second maneuver is similar to the

If the surface of the unitary-radius cylinder of gear Gi is fi rst one, but for the reliance on the contact between left-hand now cut along the generator that intersects the negative x-axis fl anks. of reference frame Wi (Fig. 2), and subsequently fl attened The fi rst step of the fi rst maneuver consists of making

(Fig. 4), the former projections of the base helices of a tooth the reference involute helicoid Σ1,1 of gear G1 go through the of gear Gi appear as straight lines. In Figure 4, coordinate extremity P1,1 of the line of contact between helicoids of right-

δi—measured from the generator that intersects the positive x- hand ﬂ anks (Fig. 2). The corresponding rotation ∆γ01a of gear axis of Wi—parameterizes the generators of the unitary-radius G1 is given by cylinder associated with gear Gi. The common-normal angular base tooth thickness ϕ0i is also shown in Figure 4, together (19) with the point Hi of intersection of the common normal A1A2 with the unitary-radius cylinder of gear Gi. where θ1,1 and b1,1 are provided by Equations 13 and 14,

The orientations of gears G1 and G2 about their respective respectively. Equation 19 can be justifi ed by elementary axes are defi ned here by considering an arbitrarily selected geometric reasoning on the unitary-radius cylinder of gear G1. reference right-hand fl ank (involute helicoid) Σ1,1 on gear G1, (The reader is referred to Reference 16 for further details.) together with the right-hand fl ank (involute helicoid) Σ2,1 of The second step makes helicoid Σ1,1 go through point P2,1. the tooth of gear G2 in contact with Σ1,1. The reference angular The necessary rotation of gear G1 is continued position of gear Gi is chosen here as characterized by helicoid www.geartechnology.com July 2008 GEARTECHNOLOGY 0059 (20) (27)

The angular position of gear G1 at the end of the whole maneuver is provided by Now helicoids Σ1,1 and Σ2,1 touch each other at P2,1. By the third—and last—step, gear G2 is so turned about its axis as (28) to make the base helix of helicoid Σ2,1 intersect the common normal A1A2. The corresponding rotation of gear G2 is given by Therefore in addition to Equation 23, another constraint

between γ01 and γ02 can be found (21) (29) At the end of the considered three-step maneuver, gear By considering the expressions of ∆ ' and ∆ " provided by G2 is at its reference position (γ02 = 0), whereas gear G1 is at a γ 01 γ 01 position identiﬁ ed by Equations 22 and 28, Equations 23 and 29 can be rewritten as

(30) (22) (31) Owing to Equation 18, and to the existence of a meshing Quantities B' and B" that appear in these equations are conﬁ guration characterized by (γ01, γ02 ) = (∆γ′01, 0) , the given by ensuing relation between γ01 and γ02 must be satisﬁ ed

(23) (32)

Now the second maneuver is taken into account. Its ﬁ rst step consists of bringing gear G1 from the reference angular (33) position to the position where helicoid Σ1,–1 goes through point

P1,–1. Here Σ1,–1 is the helicoid that, together with the reference helicoid Σ1,1 deﬁ ned above, bounds the same tooth of gear G1. For a given backlash-free beveloid gear set, Equations 30 The corresponding rotation of gear G1 is (see Figs. 2–4): and 31 must be satisﬁ ed simultaneously. Since the considered (24) gear set is a mechanism with one degree of freedom, it should

be possible to arbitrarily select γ01 (or γ02), and then determine (or ). Therefore Equations 30 and 31, when considered By the second step, gear G1 is revolved until helicoid Σ1,–1 γ02 γ01 as linear equations in and , should be linearly dependent. goes through point P2,–1. The incremental rotation of gear G1 γ01 γ02 is provided by This requirement translates into the ensuing condition:

(25) (34)

By taking into account Equations 32 and 33, Equation 34 After this step, the helicoid Σ1,–1 of gear G1 touches the can be rewritten as follows (Ref. 16): helicoid Σ2,–1 of gear G2 at point P2,–1. It is worth observing that, while Σ1,1 and Σ1,–1 bound the same tooth of gear G1, Σ2,1 (35) and Σ2,–1 delimit the same tooth space of gear G2. The third step of the current maneuver consists in making where helicoid Σ2,–1 intersect the common normal A1 A2 at a point of its base helix. The additional rotation of gear G2 is provided by (36)

(26) and

(37) The fourth—and last—step brings gear G2 at the reference position γ02 = 0, i.e., makes helicoid Σ2,1 intersect the common Equivalent to the equation set composed of Equations normal A1A2 at a point of the base helix. The corresponding 30 and 31, the equation set formed by Equations 30 and 35 further rotation of gear G2 is 6000 GEARTECHNOLOGY July 2008 www.geartechnology.com P 3–i, –1 P3–i,1 encapsulates the meshing condition—with no backlash—of a pair of conical involute gears, each bound to revolve about xi its own axis. More speciﬁ cally, Equation 35 involves the �i geometry of the two beveloid gears, together with the relative placement of the two gear axes and the axial placement of Pi, –1 Pi,1 � each gear on its own axis. It had already been presented, �i i,1 �i, –1 � � i though in a slightly different form, in Reference 16. On the � 0i other hand, by also encompassing the angular position of i Hi

the two gears, Equation 30 provides information about their �0i –1 Ai phasing. To this author’s knowledge, no such equation has tan ki,–1 yi –1 ever been published before. tan ki,1 Unconstrained Beveloid Gears in Tight Mesh � �i,1 i, –1 As anticipated at the beginning of the previous section, the zi whole collection of relative rigid-body positions of two tightly meshing beveloid gears can be determined by generalizing the results just found for a conventional beveloid gear set. Figure 5—Reference and common-normal parameters. Figure 3 . Coordinates of the extremities of the lines oLetf con usta ct consider. again a beveloid gear set, composed of two meshing beveloid gears connected to a rigid frame A1A2, and the radial line through the point on the base helix through revolute pairs. If the distance a and angle 0 α0 of reference ﬂ ank Σi,1. As shown in Figure 5, the following between the revolute pair axes, together with the geometry relation exists between γ0i and γi of the two gears—notably their common-normal base angular

thicknesses ϕ01 and ϕ02—comply with Equation 35, then (39) Equation 30 is satisﬁ ed by a simple inﬁ nity of values for the

ordered pair (γ01, γ02), i.e., the mechanism has a simple inﬁ nity By taking into account Equations 38 and 39, Equations of conﬁ gurations. 35 and 30 are transformed into a set of two equations in the Now the two mentioned revolute pairs are replaced by four unknowns ζ1, ζ2, γ1 and γ2. It is generally possible to cylindrical pairs, which implies that both gears can be displaced arbitrarily choose either of γi (i=1,2) and either of ζi (i=1,2), along their axes, in addition to being revolved about them. and then determine the remaining two unknowns. Therefore a

Consequently, the common-normal base angular thickness ϕ0i set of two beveloid gears connected to the frame by cylindrical

of gear Gi, i.e., the base angular thickness on a cross section pairs has two degrees of freedom, with no need to satisfy any

of gear Gi going through point Ai (Fig. 2), becomes linearly prerequisite similar to Equation 35. dependent on the axial placement of the gear. With the aid of At this point, the frame of the gear set is suppressed Figure 5, the ensuing condition can be laid down (See also altogether, so that parameters a0 and α0 are no longer bound to Eq. 16) be constant. We are now left with two beveloid gears that can be freely moved in space, provided that they keep meshing (38) with no backlash. All relative rigid-body positions of the two gears are those satisfying Equations 35 and 30, which can be

In this equation, ϕi is the reference base angular thickness rewritten in the ensuing concise form

of gear Gi, i.e., the base angular thickness of a tooth of gear Gi

at a reference cross section ﬁ xed to the gear. Moreover, ζi is the displacement of the common-normal cross section, measured (40) from the reference cross section, positive if concordant with

the direction of the z-axis of reference frame Wi (a positive ζi is shown in Figure 5). In Equation 40, U and V are, respectively, what the left- Quantity is no longer suited to parameterize the angular γ0i hand sides of Equations 35 and 30 turn into, once ϕ0i and γ0i

position of gear Gi. For instance, if γ0i = 0 and gear Gi is axially (i=1,2) have been replaced with the expressions provided by

displaced, then the base helix of the reference helicoid Σi,1 Equations 38 and 39.

keeps intersecting the minimum distance segment A1A2, which Equation 40 is a set of two conditions in six unknowns, means that the gear undergoes a screw motion with respect to namely, a0, α0, ζ1, ζ2, γ1 and γ2. Therefore, Equation 40 can be the rigid gear-set frame, thus changing its orientation. solved in a quadruple inﬁ nity of ways, which means that there

Henceforth the angular position of gear Gi will be is a quadruple inﬁ nity of possible relative placements for the

parameterized by angle γi, which is an angle measured on the two beveloid gears.

reference cross section of the gear. Precisely, γi is the angle As explained hereafter, each of these relative placements between two lines belonging to the reference cross section can be determined by relying on the values of a0, α0, ζ1, ζ2, γ1 continued of gear Gi—the projection of the minimum distance segment www.geartechnology.com July 2008 GEARTECHNOLOGY 0061 and γ . At fi rst, a skeleton is built based on a and α . Such a 2 0 0 (41 skeleton is formed by the axes of the two gears, together with the common normal segment A1A2 (Fig. 2). Subsequently gear G (i= 1,2) is axially placed on its axis by relying on the value of i The differentials on the right-hand sides of Equation 41 ζ . Finally, the orientation of gear G about its axis—and with 1 i are computed with these assumptions respect to the skeleton segment A1A2—is provided by angle γi. Equation 40 will prove pivotal in computing the hobbing parameters of a beveloid gear. (42) Gear-Hob Relative Movements As already mentioned at the beginning of the section addressing a backlash-free beveloid gear set, the tooth fl anks Ratio ƒ is the rate of change of the gear-hob axis distance of a beveloid gear result from a two-parameter envelope a as the hob is moved along the gear width; it is zero for by the hob thread fl anks. The double-infi nite subset of the cylindrical gears, but not so for beveloid gears. Ratio ƒ , on quadruple infi nity of possible gear-hob relative placements γ the other hand, provides information about the hob rotation is chosen here on the analogy of the hobbing operation of a as the hob is fed across the work. It would be zero for a spur cylindrical gear by a cylindrical hob. Throughout this section, gear, but is different from zero for helical gears and for most the generated gear and the enveloping hob will be referred to beveloid gears. as gear G and G respectively. 1 2 To compute ratios ƒ and ƒ , Equation 40 is now As is known, a cylindrical gear can be hobbed by keeping a γ differentiated by taking into account Equation 42 constant the work-hob axis angle α0, by revolving the hob about its axis, and by simultaneously moving such an axis across the gear width. In case no hob shift takes place during (43) hobbing—as usually happens—parameter ζ2 is kept constant too. If the gear and the hob are both cylindrical, it is easy to prove that function U in Equation 40 is deprived of arguments Based on Equation 43, the ensuing expression for quantities ζ1 and ζ2. Therefore, the constancy of the shaft angle α0 ƒa and ƒγ can be obtained implies the constancy of the axis distance a0, too; parameters γ and ζ can be thought of as the two parameters of the 1 1 (44) enveloping process. And for any choice of their values, the second condition in Equation 40 yields quantity γ2. Now the hobbing of a beveloid gear by a beveloid hob is (45) analyzed. Similar to the previous case, the shaft angle α0 is supposed as constant. Its value might be chosen, for instance, The partial derivatives in Equations 44 and 45 can be with the aim of minimizing the shaft axis distance a at a given 0 easily computed as shown hereafter. The partial derivative of cross-section of the gear (see, for instance, Reference 17 for U with respect to a is given by the ensuing concatenation of application of this criterion to the hobbing of cylindrical 0 relations gears). The independent parameters of the envelope are again γ and ζ , whereas parameter ζ is kept constant. For any 1 1 2 (46) choice of γ1 and ζ1, the fi rst and second conditions in Equation

40 yield, respectively, the values of a0 and γ2. The instantaneous movement of the hob relative to the (47) gear being machined can be thought of as the superimposition of two movements: 1. The relative movement of hob and gear as they revolve (48) about their axes (only the independent envelope parameter γ1 varies) 2. The relative movement of hob and gear when the gear (49) is shifted along its axis without turning with respect to the hobbing machine (only the independent envelope parameter ζ varies) 1 (50) Since the former movement can be found straightforwardly via Equation 18, only determination of the latter will be pursued As for the partial derivative of U with respect to ζ , it is here. More speciﬁ cally, the ensuing ratios of differentials are 1 provided by of interest

6200 GEARTECHNOLOGY July 2008 www.geartechnology.com considered as involute cylindrical gears rather than beveloid (51) gears, the ensuing additional conditions come into play

(62) (52)

The values of ratios ƒa and ƒγ, are always needed when (53) programming a CNC hobbing machine that has to cut a beveloid gear. Thanks to Equations 57 and 58, these ratios can

The partial derivative of V with respect to a0 is given by be straightforwardly assessed. Therefore, from a utilitarian standpoint, Equations 57 and 58 are the main result of this (54) paper.

Should angle α0 change while ζ1 varies (for instance, to The derivatives on the right-hand side of this equation are in constantly minimize the shaft distance a0), and/or hob shifting turn provided by Equations 49 and 50. occur during machining (for instance, to reduce the scalloping

Finally, the partial derivative of V with respect to ζ1 and γ2 of the tooth ﬂ anks of the gear), then the mentioned equations can be easily determined based on Equations 30 and 39: would no longer be applicable. In this occurrence, Equation 40—the true theoretical contribution of this paper—should be (55) resorted to again, and applied afresh to the case at hand. Numerical Example (56) The formulae derived in the previous section are here

applied to determine quantities ƒa and ƒγ for the hobbing of a

beveloid gear (gear G1) by a cylindrical hob (gear G2). Thanks to Equations 46–55, the ratios fa and fγ can be re written Throughout this section, non-integer quantities are in the ensuing explicit form: expressed by a high number of digits—all meaningful—in order to allow the reader to accurately check the reported (57) result.

The hob has one thread (N2 = 1) and is characterized by a module of 2 mm and a pressure angle of 20°. Based on this (58) data, the ensuing dimensions can be easily computed

ρ2,1= ρ2,–1= 2.735229431127 mm Quantities Dj (j= ± 1) in Equations 57 and 58 are deﬁ ned by:

β2,1= β2,–1= 69.90659103379 deg (59)

ϕ2 = 1220.353746100 deg Equations 57–59 show that ƒa and ƒγ, do not depend on ζ1, but only on the geometry of gear and hob, together with the The normal base pitches of hob and gear can be derived swivel angle α0 (which has been supposed as constant). The from the geometry of the hob constancy of ƒa, in particular, explains why the root surface of a beveloid gear is conical—at least if the gear is cut by p1= p-1= 5.904262868187 mm keeping constant the angle α0.

Equation 59 provides the expression of Dj (j= ± 1) for any The gear is characterized by pair of beveloid gears. When gear G2 is a hob, the number of teeth N is small and the absolute values of the base helix 2 N1=14 angles β2,1 and β2,−1 are relatively large. Therefore the ensuing inequality is generally satisﬁ ed ρ1,1 = 13.16838183156 mm

(60) ρ1,-1 = 13.15931278723 mm

Consequently, Equation 10 reduces to β1,1 = –2.515092347823 deg

(61) β1,-1 = –1.343231785965 deg

In addition, since the most commonly used hobs can be On a given (reference) cross section of the gear, the

www.geartechnology.com July 2008 GEARTECHNOLOGY 6300 angular base thickness of the gear teeth is: 5. Mitome, K. “Conical Involute Gear (Part 3—Tooth Action of a Pair of Gears),” Bulletin of the JSME, 28, No. 245, pp.

ϕ1= 16.94800000000 deg 2757–2764. 6. Liu C.-C. and C.-B. Tsay. “Contact Characteristics of

The angle α0 between the axes of gear and hob is chosen Beveloid Gears,” 2002, Mechanism and Machine Theory, 37, in such a way as to minimize the shaft distance a0 when point pp. 333–350.

A1 (Fig. 2) lies on the mentioned reference cross section of the 7. Börner, J., K. Humm and F.J. Joachim. “Development gear. The corresponding values of α0 and a0 are: of Conical Involute Gears (Beveloids) for Vehicle Transmissions,” Gear Technology, Nov./Dec., 2005,

α0 = –86.03648170877 deg pp. 28–35. 8. Liu C.-C. and C.-B. Tsay. “Tooth Undercutting of Beveloid

a0 = 43.96806431329 mm Gears,” 2001, ASME Journal of Mechanical Design, 123, pp. 569–576.

The ratio fa between the change of axis distance a0 and the 9. Brauer, J. “Analytical Geometry of Straight Conical displacement ζ1 of point A1 (Figs. 2 and 5) along the gear axis Involute Gears,” 2002, Mechanism and Machine Theory, 37, is provided by Equation 57 pp. 127–141. 10. Mitome, K. “Conical Involute Gear—Analysis, Design,

ƒa = 0.02988732132842 mm/mm Production and Applications,” 1981, Int. Symposium on Gearing & Power Transmissions, Tokyo, pp. 69–74.

On the other hand, the ratio ƒγ between the rotation angle 11. Mitome, K. “Table Sliding Taper Hobbing of Conical of the hob and the displacement ζ1 of point A1 along the Gear Using Cylindrical Hob. Part 1—Theoretical Analysis gear axis—when the gear does not rotate with respect to the of Table Sliding Taper Hobbing,” 1981, ASME Journal of hobbing machine—is provided by Equation 58 Engineering for Industry, 103, pp. 446–451. 12. Mitome, K. “Table Sliding Taper Hobbing of Conical Gear

ƒγ = 2.050765894724 deg/mm Using Cylindrical Hob. Part 2—Hobbing of Conical Involute Gear,” 1981, ASME Journal of Engineering for Industry, 103, (To obtain this value, a conversion of unit of measurement pp. 452-455. 13. Mitome, K. “Conical Involute Gear (Part 1—Design and has been necessary, since the ratio ƒγ yielded by Equation 58 is expressed in radians per unit of length.) Production System),” 1983, Bulletin of the JSME, 26, No. 212, pp. 299–305. Conclusions 14. Mitome, K. “Inclining Work-Arbor Taper Hobbing of With reference to the generation of a beveloid gear by a Conical Gear Using Cylindrical Hob,” 1986, ASME Journal hobbing machine, the paper has presented a simple and general of Mechanisms, Transmissions, and Automation in Design, method to determine the rate of change of the hob-work axis 108, pp. 135–141. distance and the differential rotation of the hob as the hob itself 15. Mitome, K. “A New Type of Master Gears of Hard is fed across the work. Because it relies on a very few intrinsic Gear Finisher Using Conical Involute Gear,” 1991, Proc. of dimensions of a beveloid gear, the method is conducive to the Eight World Congress on the Theory of Machines and concise expressions for the desired quantities. Mechanisms, Prague, Aug. 26–31, pp. 597–600. The results presented here refer primarily to the hobbing 16. Innocenti, C. “Analysis of Meshing of Beveloid Gears,” of a beveloid gear by a beveloid hob, provided that the swivel 1997, Mechanism and Machine Theory, 32, pp. 363–373. angle remains constant. On the one hand, they can be readily 17. Innocenti, C. “Optimal Choice of the Shaft Angle for specialized to the case of a cylindrical hob cutting a beveloid Involute Gear Hobbing,” 2008, ASME Journal of Mechanical gear. On the other hand, it is easy to extend them to make Design, 130, pp. 044502.1-044502.5. provision for hob shifting and swivel angle change during hobbing.

References 1. Merritt, H.E. Gears, 1954, 3rd Edition, Sir Isaac Pitman & Carlo Innocenti is a professor of Mechanics of Machines at the Sons, London. University of Modena and Reggio Emilia, Italy. After obtaining a degree in mechanical engineering in 1984, he worked for two 2. Beam, A.S. “Beveloid Gearing,” 1954, Machine Design, different companies—as a computation engineer and as a 26, pp. 36-49. mechanical designer, respectively. He then joined academia 3. Smith, L.J. “The Involute Helicoid and the Universal Gear,” in 1990. The papers that he has authored or co-authored are mainly focused on the kinematic analysis and synthesis of planar Gear Technology, Nov/Dec., 1990, pp. 18–27. and spatial mechanisms; rotordynamics; vehicle dynamics; robot 4. Townsend, D.P. Dudley’s Gear Handbook, 1991, McGraw- calibration; gears and gearing systems. Hill. 6400 GEARTECHNOLOGY July 2008 www.geartechnology.com